Chapter 13 Maxwell’s Equations And Electromagnetic Waves

Chapter 13Maxwell’s Equations and Electromagnetic Waves13.1 The Displacement Current . 13-313.2 Gauss’s Law for Magnetism . 13-513.3 Maxwell’s Equations . 13-513.4 Plane Electromagnetic Waves . 13-713.4.1 One-Dimensional Wave Equation . 13-1013.5 Standing Electromagnetic Waves . 13-1313.6 Poynting Vector . 13-15Example 13.1: Solar Constant. 13-17Example 13.2: Intensity of a Standing Wave. 13-1913.6.1 Energy Transport . 13-1913.7 Momentum and Radiation Pressure. 13-2213.8 Production of Electromagnetic Waves . 13-23Animation 13.1: Electric Dipole Radiation 1. 13-25Animation 13.2: Electric Dipole Radiation 2. 13-25Animation 13.3: Radiation From a Quarter-Wave Antenna . 13-2613.8.1 Plane Waves. 13-2613.8.2 Sinusoidal Electromagnetic Wave . 13-3113.9 Summary. 13-3313.10 Appendix: Reflection of Electromagnetic Waves at Conducting Surfaces . 13-3513.11 Problem-Solving Strategy: Traveling Electromagnetic Waves . 13-3913.12 Solved Problems . 13-4113.12.113.12.213.12.313.12.4Plane Electromagnetic Wave . 13-41One-Dimensional Wave Equation . 13-42Poynting Vector of a Charging Capacitor. 13-43Poynting Vector of a Conductor . 13-4513.13 Conceptual Questions . 13-4613.14 Additional Problems . 13-4713.14.1 Solar Sailing. 13-4713-1

13.14.2 Reflections of True Love . 13-4713.14.3 Coaxial Cable and Power Flow. 13-4713.14.4 Superposition of Electromagnetic Waves. 13-4813.14.5 Sinusoidal Electromagnetic Wave . 13-4813.14.6 Radiation Pressure of Electromagnetic Wave. 13-4913.14.7 Energy of Electromagnetic Waves. 13-4913.14.8 Wave Equation. 13-5013.14.9 Electromagnetic Plane Wave . 13-5013.14.10 Sinusoidal Electromagnetic Wave . 13-5013-2

Maxwell’s Equations and Electromagnetic Waves13.1 The Displacement CurrentIn Chapter 9, we learned that if a current-carrying wire possesses certain symmetry, themagnetic field can be obtained by using Ampere’s law: B d s µ I0 enc(13.1.1)The equation states that the line integral of a magnetic field around an arbitrary closedloop is equal to µ0 I enc , where I enc is the conduction current passing through the surfacebound by the closed path. In addition, we also learned in Chapter 10 that, as aconsequence of the Faraday’s law of induction, a changing magnetic field can produce anelectric field, according tod E d s dt B dA(13.1.2)SOne might then wonder whether or not the converse could be true, namely, a changingelectric field produces a magnetic field. If so, then the right-hand side of Eq. (13.1.1) willhave to be modified to reflect such “symmetry” between E and B .To see how magnetic fields can be created by a time-varying electric field, consider acapacitor which is being charged. During the charging process, the electric field strengthincreases with time as more charge is accumulated on the plates. The conduction currentthat carries the charges also produces a magnetic field. In order to apply Ampere’s law tocalculate this field, let us choose curve C shown in Figure 13.1.1 to be the Amperian loop.Figure 13.1.1 Surfaces S1 and S2 bound by curve C.13-3

If the surface bounded by the path is the flat surface S1 , then the enclosed currentis I enc I . On the other hand, if we choose S2 to be the surface bounded by the curve,then I enc 0 since no current passes through S 2 . Thus, we see that there exists anambiguity in choosing the appropriate surface bounded by the curve C.Maxwell showed that the ambiguity can be resolved by adding to the right-hand side ofthe Ampere’s law an extra termdΦE(13.1.3)Id ε0dtwhich he called the “displacement current.” The term involves a change in electric flux.The generalized Ampere’s (or the Ampere-Maxwell) law now reads B d s µ I µ ε00 0dΦE µ0 ( I I d )dt(13.1.4)The origin of the displacement current can be understood as follows:Figure 13.1.2 Displacement through S2In Figure 13.1.2, the electric flux which passes through S2 is given byΦE Q E dA EA εS(13.1.5)0where A is the area of the capacitor plates. From Eq. (13.1.3), we readily see that thedisplacement current I d is related to the rate of increase of charge on the plate byId ε0d Φ E dQ dtdt(13.1.6)However, the right-hand-side of the expression, dQ / dt , is simply equal to the conductioncurrent, I . Thus, we conclude that the conduction current that passes through S1 is13-4

precisely equal to the displacement current that passes through S2, namely I I d . Withthe Ampere-Maxwell law, the ambiguity in choosing the surface bound by the Amperianloop is removed.13.2 Gauss’s Law for MagnetismWe have seen that Gauss’s law for electrostatics states that the electric flux through aclosed surface is proportional to the charge enclosed (Figure 13.2.1a). The electric fieldlines originate from the positive charge (source) and terminate at the negative charge(sink). One would then be tempted to write down the magnetic equivalent asΦB Qm B dA µS(13.2.1)0where Qm is the magnetic charge (monopole) enclosed by the Gaussian surface. However,despite intense search effort, no isolated magnetic monopole has ever been observed.Hence, Qm 0 and Gauss’s law for magnetism becomesΦB B dA 0(13.2.2)SFigure 13.2.1 Gauss’s law for (a) electrostatics, and (b) magnetism.This implies that the number of magnetic field lines entering a closed surface is equal tothe number of field lines leaving the surface. That is, there is no source or sink. Inaddition, the lines must be continuous with no starting or end points. In fact, as shown inFigure 13.2.1(b) for a bar magnet, the field lines that emanate from the north pole to thesouth pole outside the magnet return within the magnet and form a closed loop.13.3 Maxwell’s EquationsWe now have four equations which form the foundation of electromagnetic phenomena:13-5

LawEquationPhysical InterpretationQ E dA εGauss's law for E0Electric flux through a closed surfaceis proportional to the charged encloseddΦBdtChanging magnetic flux produces anelectric field B dA 0The total magnetic flux through aclosed surface is zeroSFaraday's law E d s Gauss's law for BAmpere Maxwell lawS B d s µ I µ ε00 0dΦEdtElectric current and changing electricflux produces a magnetic fieldCollectively they are known as Maxwell’s equations. The above equations may also bewritten in differential forms as E ρε0 E B t(13.3.1) B 0 B µ 0 J µ 0ε 0 E twhere ρ and J are the free charge and the conduction current densities, respectively. Inthe absence of sources where Q 0, I 0 , the above equations become E dA 0S E d s dΦBdt B dA 0(13.3.2)S B d s µ ε0 0dΦEdtAn important consequence of Maxwell’s equations, as we shall see below, is theprediction of the existence of electromagnetic waves that travel with speed of lightc 1/ µ 0ε 0 . The reason is due to the fact that a changing electric field produces amagnetic field and vice versa, and the coupling between the two fields leads to thegeneration of electromagnetic waves. The prediction was confirmed by H. Hertz in 1887.13-6

13.4 Plane Electromagnetic WavesTo examine the properties of the electromagnetic waves, let’s consider for simplicity anelectromagnetic wave propagating in the x-direction, with the electric field E pointingin the y-direction and the magnetic field B in the z-direction, as shown in Figure 13.4.1below.Figure 13.4.1 A plane electromagnetic waveWhat we have here is an example of a plane wave since at any instant both E and B areuniform over any plane perpendicular to the direction of propagation. In addition, thewave is transverse because both fields are perpendicular to the direction of propagation,which points in the direction of the cross product E B .Using Maxwell’s equations, we may obtain the relationship between the magnitudes ofthe fields. To see this, consider a rectangular loop which lies in the xy plane, with the leftside of the loop at x and the right at x x . The bottom side of the loop is located at y ,and the top side of the loop is located at y y , as shown in Figure 13.4.2. Let the unitvector normal to the loop be in the positive z-direction, nˆ kˆ .Figure 13.4.2 Spatial variation of the electric field EUsing Faraday’s lawd E d s dt B dA(13.4.1)13-7

the left-hand-side can be written as E ds E ( x x) y E ( x) y [ E ( x x) E ( x)] y yyyy E y x( x y)(13.4.2)where we have made the expansionE y ( x x) E y ( x) E y x x (13.4.3)On the other hand, the rate of change of magnetic flux on the right-hand-side is given by d B B dA z ( x y ) dt t (13.4.4)Equating the two expressions and dividing through by the area x y yields E y x Bz t(13.4.5)The second condition on the relationship between the electric and magnetic fields may bededuced by using the Ampere-Maxwell equation: B ds µ ε0 0dE dAdt (13.4.6)Consider a rectangular loop in the xz plane depicted in Figure 13.4.3, with a unit normalnˆ ˆj .Figure 13.4.3 Spatial variation of the magnetic field BThe line integral of the magnetic field is13-8

B d s B ( x) z B ( x x) z [ B ( x) B ( x x)] zzzzz(13.4.7) B z ( x z ) x On the other hand, the time derivative of the electric flux isµ 0ε 0 Ey dE d A µ 0ε 0 ( x z ) dt t (13.4.8)Equating the two equations and dividing by x z , we have Ey Bz µ 0ε 0 x t (13.4.9)The result indicates that a time-varying electric field is generated by a spatially varyingmagnetic field.Using Eqs. (13.4.4) and (13.4.8), one may verify that both the electric and magnetic fieldssatisfy the one-dimensional wave equation.To show this, we first take another partial derivative of Eq. (13.4.5) with respect to x, andthen another partial derivative of Eq. (13.4.9) with respect to t: 2 Ey x 2 E y 2 Ey Bz Bz µεµε 0 0 0 0 x t t x t t t 2(13.4.10)noting the interchangeability of the partial differentiations: Bz Bz x t t x (13.4.11)Similarly, taking another partial derivative of Eq. (13.4.9) with respect to x yields, andthen another partial derivative of Eq. (13.4.5) with respect to t gives E y 2 Bz E y Bz µ 0ε 0 µ 0ε 0 µ 0ε 0 2 x x t t x t t 2 Bz µε 0 0 t 2 (13.4.12)The results may be summarized as: 2 2 E y ( x, t ) µε 0 2 0 0 t 2 Bz ( x, t ) x(13.4.13)13-9

Recall that the general form of a one-dimensional wave equation is given by 21 2 ψ ( x, t ) 0 222 x v t (13.4.14)where v is the speed of propagation and ψ ( x, t ) is the wave function, we see clearly thatboth E y and Bz satisfy the wave equation and propagate with the speed of light:v 1µ 0ε 0 1(4π 10 T m/A)(8.85 10 7 12C /N m )22 2.997 108 m/s c(13.4.15)Thus, we conclude that light is an electromagnetic wave. The spectrum ofelectromagnetic waves is shown in Figure 13.4.4.Figure 13.4.4 Electromagnetic spectrum13.4.1 One-Dimensional Wave EquationIt is straightforward to verify that any function of the form ψ ( x vt ) satisfies the onedimensional wave equation shown in Eq. (13.4.14). The proof proceeds as follows:Let x′ x vt which yields x′ / x 1 and x′ / t v . Using chain rule, the first twopartial derivatives with respect to x are ψ ( x′) ψ x′ ψ x x′ x x′(13.4.16)13-10

2ψ ψ 2ψ x′ 2ψ x 2 x x′ x′2 x x′2(13.4.17)Similarly, the partial derivatives in t are given by ψ ψ x′ ψ v t x′ t x′(13.4.18)2 2ψ ψ 2ψ x′2 ψvvv t 2 t x′ x′2 t x′2(13.4.19)Comparing Eq. (13.4.17) with Eq. (13.4.19), we have

Maxwell’s Equations and Electromagnetic Waves 13.1 The Displacement Current In Chapter 9, we learned that if a current-carrying wire possesses certain symmetry, the